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Symplectic nonsqueezing for the KdV flow on the line

Maria Ntekoume
Vol. 4 (2022), No. 3, 401–448
DOI: 10.2140/paa.2022.4.401
Abstract

We show symplectic nonsqueezing for the KdV equation on the line ℝ. This is achieved via finite-dimensional approximation. Our choice of finite-dimensional Hamiltonian system that effectively approximates the KdV flow is inspired by the recent breakthrough of Killip and Vişan (Ann. of Math. (2) 190:1 (2019), 249–305) in the well-posedness theory of the equation in low-regularity spaces, relying on its completely integrable structure. The employment of our methods also provides us with a new concise proof of symplectic nonsqueezing for the same equation on the circle 𝕋, recovering the result of Colliander et al. (Acta Math. 195 (2005), 197–252).

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Keywords
KdV, symplectic nonsqueezing, finite-dimensional approximation
Mathematical Subject Classification
Primary: 35Q53
Milestones
Received: 17 July 2020
Revised: 19 January 2022
Accepted: 10 March 2022
Published: 4 December 2022
Authors
Maria Ntekoume
Department of Mathematics
Rice University
Houston, TX
United States